Now there is an alternative way to get the canonical sequence directly from the augmented adjacency matrix. Let \(G=(V,E)\)  be a proper interval graph with \(V=\left\lbrace v_i\ \vert \ i=1,2,\ldots ,n\right\rbrace \)  and \(A(G)\)  be the augmented adjacency matrix of \(G\)  with consecutive 1's property. We partition positions of \(A(G)\)  into two sets \((L,U)\)  by drawing a polygonal path from the upper left corner to the lower right corner such that the set \(L\)  [resp. \(U\) ] is closed under leftward or downward [respectively, rightward or upward] movement (called a stair partition [1]}, [2]}, [3]}) and \(U\)  contains precisely all the zeros right to the principal diagonal of \(A(G)\)  (see Table REF ). This is possible due to the consecutive 1's property of \(A(G)\) . Now we obtain a sequence of positive integers belonging to \(\left\lbrace 1,2,\ldots ,n\right\rbrace \) , each occurs exactly twice, by writing the row or column numbers as they appear along the stair. We call this sequence, the stair sequence of \(A(G)\)  (see Table REF ) and note that it is same as the canonical sequence of \(G\)  with respect to the given canonical ordering of vertices of \(G\) .
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